Theorem eqelssd 3198

Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
eqelssd.1	|- ( ph -> A C_ B )
eqelssd.2	|- ( ( ph /\ x e. B ) -> x e. A )

Assertion

Ref	Expression
eqelssd	|- ( ph -> A = B )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem eqelssd

Step	Hyp	Ref	Expression
1		eqelssd.1	. 2 |- ( ph -> A C_ B )
2		eqelssd.2	. . . 4 |- ( ( ph /\ x e. B ) -> x e. A )
3	2	ex 115	. . 3 |- ( ph -> ( x e. B -> x e. A ) )
4	3	ssrdv 3185	. 2 |- ( ph -> B C_ A )
5	1, 4	eqssd 3196	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   C_ wss 3153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  fiuni  7037  ennnfonelemrn  12576  ennnfonelemdm  12577  unirnblps  14590  unirnbl  14591  dvidlemap  14845  dviaddf  14854  dvimulf  14855  dvcj  14858  dvrecap  14862